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Properties

Label 2-1100-55.4-c1-0-6
Degree $2$
Conductor $1100$
Sign $0.320 - 0.947i$
Analytic cond. $8.78354$
Root an. cond. $2.96370$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.151 + 0.0492i)3-s + (−1.93 + 0.628i)7-s + (−2.40 − 1.74i)9-s + (2.88 + 1.63i)11-s + (−0.384 + 0.528i)13-s + (2.52 + 3.47i)17-s + (0.919 − 2.82i)19-s − 0.324·21-s + 6.11i·23-s + (−0.559 − 0.770i)27-s + (2.63 + 8.11i)29-s + (4.34 + 3.15i)31-s + (0.357 + 0.389i)33-s + (−3.28 + 1.06i)37-s + (−0.0843 + 0.0612i)39-s + ⋯
L(s)  = 1  + (0.0875 + 0.0284i)3-s + (−0.731 + 0.237i)7-s + (−0.802 − 0.582i)9-s + (0.870 + 0.492i)11-s + (−0.106 + 0.146i)13-s + (0.612 + 0.843i)17-s + (0.210 − 0.649i)19-s − 0.0708·21-s + 1.27i·23-s + (−0.107 − 0.148i)27-s + (0.489 + 1.50i)29-s + (0.780 + 0.567i)31-s + (0.0622 + 0.0678i)33-s + (−0.540 + 0.175i)37-s + (−0.0134 + 0.00980i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.287535350\)
\(L(\frac12)\) \(\approx\) \(1.287535350\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + (-2.88 - 1.63i)T \)
good3 \( 1 + (-0.151 - 0.0492i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.93 - 0.628i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.384 - 0.528i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.52 - 3.47i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.919 + 2.82i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.11iT - 23T^{2} \)
29 \( 1 + (-2.63 - 8.11i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.34 - 3.15i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.28 - 1.06i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.47 + 4.53i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (-6.94 - 2.25i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.53 - 8.99i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.70 + 5.24i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-7.77 + 5.64i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.60iT - 67T^{2} \)
71 \( 1 + (0.0442 - 0.0321i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.78 - 2.20i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.37 - 1.72i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.25 + 9.98i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.00487T + 89T^{2} \)
97 \( 1 + (10.7 - 14.7i)T + (-29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.795586355369149215741290075620, −9.240017956252944615584756944464, −8.575943188820311522775710300553, −7.46342977767049513090816107466, −6.56700328388650369729293164589, −5.93875388231027215972275861976, −4.85386047636054773540423104868, −3.62181558279128363125791730473, −2.94785858244070016772457318457, −1.35799894403851293058557352023, 0.60622658304716839612024982453, 2.40547602522024944569772199486, 3.33821494920828442366848636194, 4.36919214863017046308314193776, 5.56081271233961343205614175715, 6.26543845457092303110864537684, 7.17342767226686509074474028157, 8.163316362729138706568539141366, 8.781250904515375357787719940940, 9.818642012087489242936422310128

Graph of the $Z$-function along the critical line